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• Author or Editor: Barbara J. Reys
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## Promoting Number Sense in the Middle Grades

Phrases such as “number sense,” “Operation sense,” and “intuitive understanding of number” are used throughout the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) to describe an intangible quality possessed by successful mathematics learners. Number sense refers to an intuitive feeling for numbers and their various uses and interpretations, an appreciation for various levels of accuracy when computing, the ability to detect arithmetical errors, and a common-sense approach to using numbers (Howden 1989; McIntosh, Reys, and Reys 1991). Number sense is not a finite entity that a student either has or does not have but rather a process that develops and matures with experience and knowledge. It does not develop by chance, nor does being skilled at manipulating numbers necessarily reflect this acquaintance and familiarity with numbers. Above all, number sense is characterized by a desire to make sense of numerical situations, including relating numbers to context and analyzing the effect of manipulations on numbers. It is a way of thinking that should permeate all aspects of mathematics teaching and learning.

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## Mental Computation

The ability to compute mentally. that is. to calculate exact numerical answers without the aid of any calculating or recording device, varies tremendously among indi vidual s. A quick survey of any elementary or secondary school classroom will document that some students perform mental computation quickly and accurately, whereas others are greatly hampered on even simple arithmetic problems by the withdrawal of paper and pencil. Occasionally history reports a case of an individual who ha developed extraordinary kill in mental computation. These individuals have both fascinated us and made us curious about the development of such skill. One such example wa Zerah Colburn. Born in Vermont in 1804, the son of a farmer, by the age of eight he was touring America and England displaying his exceptional mental calculating ability. He was able to give instantly the product of any two four-digit numbers. Asked to raise 8 to the sixteenth power, he gave the correct answer of 281 474 976 710 656 in a few seconds without the aid of any recording device (Eves 1969).

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## Estimation and Mental Computation It's “About” Time

Mental computation and estimation are topics we keep hearing about. Although mental computation is not a stranger to the history of mathematics education, estimation is a relative newcomer to the curriculum. The history of mental computation dates back to when arithmetic was first taught. Historically, it was emphasized bccause of its social utility—shopkeepers needed to “cipher” total of grocery bills quickly. The current emphasis on both mental computation and estimation has a different origin. As teachers, we want students to take advantage of whatever computatonal tool is most appropriate for the situation at hand. Yet unless students have developed the skill both to compute mentally and to estimate and the awareness to take advantage or whichever is appropriate, we will not see it happen.

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## If Only Clairaut Had Dynamic Geometry Tools

Using Clairaut's historic-dynamic approach and dynamic geometry tools in middle school can develop students' conceptual understanding before they encounter formal proof in geometry.

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## Principles and Standards (2001): October 2001

### Clearing up the Confusion over Calculator Use in Grades K-5

Since the publication of NCTM's Principles and Standards for School Mathematics in April 2000, considerable discussion has taken place about “key messages” of the document. The breadth of the content of Principles and Standards may hamper attempts to identify messages about particular topics. In addition, many of the fundamental messages are not easily distilled into short phrases. In fact, when such messages are too succinctly articulated, the danger of oversimplification and misunderstanding arises. This misapprehension can be seen in a question that often emerges in discussions about elementary school mathematics and Principles and Standards. That is, what does Principles and Standards say about calculator use in elementary school?

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## In My Opinion: Japanese Mathematics Education: What Makes It Work?

Japan's stature as an economic and political power worldwide has caused growing interest in the country's culture and. more specifically, its system of educating its youth. International comparisons of mathematics achievement highlight Japanese students' unquestioned superiority in mathematical performance. Factors that contribute to the relatively high performance include the nature of Japanese schools, the professional stature of teachers, the homogeneity of the school population, the high parental expectations for the educational success of their children, the abundance of jukus (special cram schools), and heavy reliance on entrance and qualifying examinations.

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## Recruiting Mathematics Teachers: Strategies to Consider

The federal No Child Left Behind Act (Public Law 107-110, HR 1, 2001) calls for all teachers in schools receiving federal funds to be “highly qualified.” That is, they must hold a bachelor's degree, demonstrate competence in the subject matter that they teach, and have full state teacher certification—their certification requirements cannot be waived nor can they have an “emergency, provisional, or temporary” certificate. These requirements are mandatory by the 2005–2006 school year. However, a serious shortage of mathematics teachers continues to exist in middle and secondary schools. For example, the Missouri Department of Elementary and Secondary Education reported that during the past fifteen years, an average of fewer than 200 mathematics teaching certificates (middle school and secondary) were issued annually by thirty-four different colleges and universities in the state. This average is far short of the more than 400 job listings that Missouri school districts annually post for middle school and secondary mathematics teachers.

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## Computation in the Elementary Curriculum: Shifting the Emphasis

Elementary teachers receive conflicting messages about the value of various computa-tional techniques, mental and written, as well as about what strategies, invented and standard, should be introduced and developed at different levels within the elementary school curriculum. They receive advice and directives from educational specialists and national and state curriculum documents.

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## One Point of View: Mental Computation and Computational Estimation—Their Time Has Come

Changing curricula in mathematics is more difficult than moving an old graveyard in January. Nevertheless, cries for changing our mathematics programs are coming from many directions as ideas for a forward-looking, futuristic mathematics curriculum are offered. Although calls for specific curricular changes are varied, all seem to agree on one thing: mathematics programs must give significantly more attention to the development of skills in mental computation and estimation and much less attention to traditional written algorithms for computation.

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## Calculators in the Classroom: How Can We Make It Happen?

Although curriculum and teaching method change slowly, we have witnessed some major changes in mathematics instruction over the last ten years, most notably the emphasis placed on the problem-solving process. We've also observed vast amount of resources funneled into microcomputer equipment for young students. In part because of this interest in microcomputer technology, we may have overlooked a technological device that has even more potential for revitalizing and enhancing mathematics instruction for children. Simple four-function calculators have been hoved to the back burner (or rear shelf of the school storeroom) in the past few years, but interest in them is reviving as evidenced by this focus issue.